We know that $ \dfrac{2}{2n^5-3} > \dfrac{2}{2n^5} = \dfrac{1}{n^5}>0$ for any $n\ge 2$. Considering this fact, what does the direct comparison test say about $\sum\limits_{n=2}^{\infty }\dfrac{2}{2n^5-3}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A The series converges. (Choice B) B The series diverges. (Choice C) C The test is inconclusive.
$\sum_{n=1}^{\infty }{\frac{1}{{{n}^{5}}}}$ is a $p$ -series with $p=5$, so it converges. Because our given series is term-by-term greater than a convergent series, the direct comparison test does not apply. So the direct comparison test is inconclusive.